Two small spheres of each charge q , mass m and material density d are suspended from a fixed point with the help of inextensible light thread. When the spheres are in air, the angle between the threads is 90^ . When the spheres are suspended in a liquid of density 2 3 d , the angle between the threads is 60^ . The value of dielectric constant of the liquid is

Step-by-step solution

Key Idea At any instant, T =m g and T =F_e m g=F_e aligned F_e & = k q^2 r^2 r & =2 l & = F_e m g aligned [from Fig.] So, for =45^ and =45^ aligned r & = 2 l v^2 = 2 l F_e= K q^2 2 l 45^ & = k q^2 2 l m g ( 45^ =1 ) aligned For, =30^ and =60^ r=2 l 1 2 =l So , F_e^ = k q^2 l^2 30^ = k^ q^2 l^2-m^ g As, the sphere is suspended in a liquid of density 2 3 d , then the observed weight of the body, m^ =V (d- 2 d 3 )= m 3 [ m=V d] So, from Eq. (i) and (ii), we get aligned 3 3 k^ & = k 2 k^ = 1 4 ₀ _r , K= 1 4 ₀ _r & =6 3